Hard GRE Statistics Practice Questions Practice Questions

Concept Explanation

Statistics on the GRE Quantitative Reasoning section involves the collection, analysis, interpretation, and presentation of data sets using measures of central tendency, dispersion, and probability distributions. To excel at Hard GRE Statistics Practice Questions, you must move beyond simple averages and understand how data shifts impact the standard deviation and how normal distribution percentiles function in a standardized testing environment. Frequently, the test will challenge your ability to synthesize disparate concepts, such as combining weighted averages with range constraints or interpreting box plots alongside frequency tables.

Key concepts include the arithmetic mean, median, mode, and range, but high-level questions focus on standard deviation and the properties of the normal curve. Standard deviation measures how much the data points in a set vary from the mean. A crucial rule for the GRE is that if every number in a set is increased by a constant $k$, the mean increases by $k$, but the standard deviation remains unchanged. Conversely, if every number is multiplied by $k$, both the mean and the standard deviation are multiplied by $k$. For a deeper dive into these quantitative fundamentals, you may find our GRE Prep hub extremely useful.

Understanding the Normal Distribution is also vital. In a perfectly normal distribution, approximately 68% of the data falls within one standard deviation of the mean, and 95% falls within two. Harder questions often ask about the "percentile" of a specific value, requiring you to visualize the area under the curve. For more specific practice on other math areas, check out these Free GRE Practice Questions.

Solved Examples

1. Example 1: Weighted Averages
   A class of 20 students has an average test score of 82. If a group of 5 students from this class has an average score of 92, what is the average score of the remaining 15 students?
   

   1. Calculate the total sum of all scores: $20 \times 82 = 1,640$.

   2. Calculate the total sum of the 5 students' scores: $5 \times 92 = 460$.

   3. Subtract the subgroup sum from the total sum: $1,640 - 460 = 1,180$.

   4. Divide by the remaining number of students: $\frac{1,180}{15} = 78.67$.

2. Example 2: Standard Deviation Shifts
   Set $S$ consists of the integers $\{10, 20, 30, 40, 50\}$. Set $T$ is created by multiplying every element in Set $S$ by 3 and then adding 5 to each resulting number. What is the ratio of the standard deviation of Set $T$ to the standard deviation of Set $S$?
   

   1. Recall the rule: adding a constant does not change standard deviation, but multiplying by a constant $k$ multiplies the standard deviation by $|k|$.

   2. The addition of 5 has zero effect on the spread of the data.

   3. The multiplication by 3 scales the standard deviation by a factor of 3.

   4. Therefore, the ratio is $3:1$.

3. Example 3: Normal Distribution Percentiles
   In a normally distributed set of 1,000 observations with a mean of 50 and a standard deviation of 10, approximately how many observations are greater than 70?
   

   1. Identify how many standard deviations 70 is from the mean: $\frac{70 - 50}{10} = 2$ standard deviations.

   2. Recall the 68-95-99.7 rule. 95% of data is within 2 standard deviations ($\pm 20$ from the mean).

   3. This means 5% of the data lies outside this range (below 30 or above 70).

   4. Since the distribution is symmetric, half of that 5% is above 70: $2.5\%$.

   5. Calculate the number of observations: $0.025 \times 1,000 = 25$.

Practice Questions

1. Set $A$ contains 7 distinct integers. If the median of Set $A$ is 15 and the largest possible range is 40, what is the smallest possible value for the least integer in the set?

2. The average (arithmetic mean) of a set of $n$ numbers is 24. When the number 48 is added to the set, the new average is 28. What is the value of $n$?

3. A data set has a mean of 150 and a standard deviation of 20. If every value in the set is decreased by 15 and then divided by 2, what is the new standard deviation?

4. In a group of 60 people, the average age is 35. If the average age of the men is 40 and the average age of the women is 32, how many women are in the group?

5. Set $X$ has a standard deviation of $\sigma$. Set $Y$ is formed by taking each element $x$ in Set $X$ and transforming it to $2x - 10$. Express the standard deviation of Set $Y$ in terms of $\sigma$.

6. The probability of an event occurring is $p$. If the event is independent and is observed 3 times, what is the probability that it occurs at least once?

7. A normal distribution has a mean of $\mu$ and a standard deviation of $\sigma$. What percentage of the data falls between $\mu - \sigma$ and $\mu + 2\sigma$?

8. If the average of five consecutive even integers is $M$, what is the average of the next five consecutive even integers in terms of $M$?

9. The mean of 10 numbers is 12. If one number is removed, the mean of the remaining 9 numbers is 13. What was the value of the number that was removed?

Answers & Explanations

1. Answer: -9. To minimize the smallest value while keeping the median at 15 and the range at 40, we need the largest value to be as small as possible. However, the range is fixed at 40. Let the set be $\{x_1, x_2, x_3, 15, x_5, x_6, x_7\}$. The range is $x_7 - x_1 = 40$. To make $x_1$ as small as possible, we don't actually have a constraint on $x_7$ other than it must be $\geq 15$. Wait, the question asks for the smallest possible value for the least integer. If the median is 15, $x_7$ must be at least 18 (if they are distinct integers: 15, 16, 17, 18). If $x_7 = 18$, then $x_1 = 18 - 40 = -22$. But if we want the smallest $x_1$, we can make $x_7$ very large? Actually, in distinct integers, $x_7$ has no upper bound, but usually, these questions imply a fixed set. If the range is 40, $x_1 = x_7 - 40$. To minimize $x_1$, we must minimize $x_7$. The smallest $x_7$ can be is 18 (since $x_4=15, x_5=16, x_6=17, x_7=18$). Thus, $18 - 40 = -22$.

2. Answer: 5. Use the sum formula: $\text{Sum}_1 = 24n$. New sum: $24n + 48$. New average: $\frac{24n + 48}{n+1} = 28$. Solving gives $24n + 48 = 28n + 28$, so $4n = 20$, $n = 5$.

3. Answer: 10. Subtracting 15 does not change the standard deviation (it remains 20). Dividing by 2 scales the standard deviation by 2. $\frac{20}{2} = 10$.

4. Answer: 37.5 (or 38 rounded). Let $w$ be the number of women. $32w + 40(60 - w) = 35(60)$. $32w + 2400 - 40w = 2100$. $-8w = -300$. $w = 37.5$. (Note: In a real GRE question, the numbers would result in an integer).

5. Answer: $2\sigma$. The constant subtraction (-10) is ignored. The multiplier 2 doubles the standard deviation.

6. Answer: $1 - (1-p)^3$. The probability of the event NOT occurring in one trial is $1-p$. For three trials, it is $(1-p)^3$. The probability of at least once is the complement: $1 - (1-p)^3$.

7. Answer: 81.5%. Between $\mu - \sigma$ and $\mu$ is 34%. Between $\mu$ and $\mu + \sigma$ is 34%. Between $\mu + \sigma$ and $\mu + 2\sigma$ is 13.5%. Total: $34 + 34 + 13.5 = 81.5\%$.

8. Answer: $M + 10$. Consecutive even integers are $x, x+2, x+4, x+6, x+8$. The mean $M$ is the middle term $x+4$. The next five are $x+10, x+12, x+14, x+16, x+18$. The new mean is $x+14$. Since $M = x+4$, the new mean is $M + 10$.

9. Answer: 3. Original sum: $10 \times 12 = 120$. New sum: $9 \times 13 = 117$. The removed number is $120 - 117 = 3$.

1. If the standard deviation of a set of numbers is 0, which of the following must be true?

• AThe mean is 0
• BThe median is 0
• CAll numbers in the set are identical
• DThe range is 1

Frequently Asked Questions

What is the difference between standard deviation and variance?

Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance. Standard deviation is typically preferred in GRE questions because it is expressed in the same units as the original data.

How do I handle outliers in GRE statistics?

Outliers significantly affect the mean and the range but have little to no impact on the median and mode. When a question asks which measure is most "robust" or least affected by an extreme value, the answer is almost always the median.

Can standard deviation be negative?

No, standard deviation can never be negative because it is calculated using squared differences and a principal square root. The minimum possible value for standard deviation is zero, which occurs when all data points are identical.

What is a weighted average?

A weighted average accounts for the relative importance or frequency of each value in a data set rather than treating them all equally. You calculate it by multiplying each value by its weight, summing those products, and dividing by the total weight.

Are normal distribution tables provided on the GRE?

No, the GRE does not provide Z-tables, so you must memorize the 68-95-99.7 rule. Knowing these percentages allows you to estimate the area under the curve for values that are 1, 2, or 3 standard deviations from the mean.

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